Random Variable - Random Signals and Processes Primer with MATLAB - page 53

Table 2.1 Continuous variables and densities
Variable Density

8
<
:

1
x 2 x 1

Uniform in the interval
[ x 1 , x 2 ]

for x 1 x x 2 ;

f X ðxÞ¼

0 otherwise :

1
2 p

Normal (Gaussian)

e ðxmÞ 2

= 2 s 2

p

for 1<x<1 , m and s 2 are parameters f X ðxÞ¼

s

(

Exponential

l e lx for x 0 ;

f X ðxÞ¼ ,

0 otherwise :

l is a parameter

Laplacian l

2 e l jj for 1<x<1 , l is a parameter

f X ðxÞ¼

<
:

Gamma

k 1 e x=b
bGðcÞ

ðx=bÞ

for x 0 ;

f X ðxÞ¼ ,

0 otherwise :

( c ) is a Gamma function.
b and c are parameters, b >

Г

0, c > 0

8
<
:

x
s 2 e x 2

Rayleigh = 2 s 2

for x 0 ;

f X ðxÞ¼ ,

0 otherwise :

for any s > 0

(

Weibull

Kx m

for x 0 ;

f X ðxÞ¼ ,

otherwise : 0

K and m are parameters

Cauchy b=p

f x ðxÞ¼ for 1<x<1; for any a and b > 0

2

b 2 þðx aÞ

<
:

Chi-Squared

x c 1 e x= 2
2 c

for x 0 ;

GðcÞ

f X ðxÞ¼

0 otherwise :

Table 2.2 Discrete variables and densities
Uniform

<
:

1
N

for x ¼ x 1 ; :::x N ;

f X ðxÞ¼

0 otherwise :

Binomial X is the number of successes in n trials

p k q nk

n
k

PfX ¼ k ; ng¼

; p þ q ¼ 1

;

f X ðxÞ¼ X

n

PfX ¼ k ; ngdðx kÞ:

k¼ 0

Poisson X is the number of arrivals in a given time t when the arrival rate is l

PfX ¼ kg¼ ð lt Þ

k

e lt ; l> 0 ; k ¼ 0 ; 1 ; 2 ; :::;

k !

f X ðxÞ¼ 1
k¼ 0

PfX ¼ kgdðx kÞ: